Theorem isphg 30888

Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isphg.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isphg	⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem isphg

Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ph 30884	. . 3 ⊢ CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))})
2	1	elin2 4143	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}))
3		rneq 5891	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
4		isphg.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
5	3, 4	eqtr4di 2789	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
6		oveq 7373	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
7	6	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
8	7	oveq1d 7382	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦))↑2) = ((𝑛‘(𝑥𝐺𝑦))↑2))
9		oveq 7373	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(-1𝑠𝑦)) = (𝑥𝐺(-1𝑠𝑦)))
10	9	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑠𝑦))))
11	10	oveq1d 7382	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2))
12	8, 11	oveq12d 7385	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)))
13	12	eqeq1d 2738	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
14	5, 13	raleqbidv 3311	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
15	5, 14	raleqbidv 3311	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
16		oveq 7373	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (-1𝑠𝑦) = (-1𝑆𝑦))
17	16	oveq2d 7383	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥𝐺(-1𝑠𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
18	17	fveq2d 6844	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑛‘(𝑥𝐺(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑆𝑦))))
19	18	oveq1d 7382	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2))
20	19	oveq2d 7383	. . . . . 6 ⊢ (𝑠 = 𝑆 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
21	20	eqeq1d 2738	. . . . 5 ⊢ (𝑠 = 𝑆 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
22	21	2ralbidv 3201	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
23		fveq1 6839	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
24	23	oveq1d 7382	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝑥𝐺𝑦))↑2))
25		fveq1 6839	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
26	25	oveq1d 7382	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2))
27	24, 26	oveq12d 7385	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
28		fveq1 6839	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑥) = (𝑁‘𝑥))
29	28	oveq1d 7382	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥)↑2) = ((𝑁‘𝑥)↑2))
30		fveq1 6839	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑦) = (𝑁‘𝑦))
31	30	oveq1d 7382	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑦)↑2) = ((𝑁‘𝑦)↑2))
32	29, 31	oveq12d 7385	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)) = (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))
33	32	oveq2d 7383	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
34	27, 33	eqeq12d 2752	. . . . 5 ⊢ (𝑛 = 𝑁 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
35	34	2ralbidv 3201	. . . 4 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
36	15, 22, 35	eloprabg 7477	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
37	36	anbi2d 631	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → ((⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
38	2, 37	bitrid 283	1 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⟨cop 4573  ran crn 5632  ‘cfv 6498  (class class class)co 7367  {coprab 7368  1c1 11039   + caddc 11041   · cmul 11043  -cneg 11378  2c2 12236  ↑cexp 14023  NrmCVeccnv 30655  CPreHil_OLDccphlo 30883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642  df-iota 6454  df-fv 6506  df-ov 7370  df-oprab 7371  df-ph 30884

This theorem is referenced by:  cncph  30890  isph  30893  phpar  30895  hhph  31249